We are given that $\{X_n\}_{n\in\mathbb N}$ are iid $\text{Bernoulli}(1/2)$. Then let $Y_n=X_n/n^{\theta},\ \theta>0$. The question is to say whether $S_n=\sum_{i=1}^nY_i$ converges almost surely or not.
Towards a solution, I note that if $\theta>1$, then since the sum is dominated by $\sum_{i=1}^n 1/i^\theta$ which has a finite limit, $S_n$ also converges, and hence in particular, converges almost surely. But for the $0<\theta\le 1$ case, I am stuck. No bound I am considering is being strong enough to point me towards a contradiction (or towards a affirmative answer, which I have a hunch is not the case, but I might be wrong).
Any help is appreciated!